Answer:
0.0384 < 0.05, which means that we can conclude that the mean amount of time that college students spend in the shower is significantly different from 5 minutes.
Step-by-step explanation:
A facilities manager at a university reads in a research report that the mean amount of time spent in the shower by an adult is 5 minutes.
This means that the null hypothesis is ![H_{0} = 5](https://tex.z-dn.net/?f=H_%7B0%7D%20%3D%205)
He decides to collect data to see if the mean amount of time that college students spend in the shower is significantly different from 5 minutes.
This means that the alternate hypothesis is ![H_{a} \neq 5](https://tex.z-dn.net/?f=H_%7Ba%7D%20%5Cneq%205)
The test statistic is:
![z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
In which X is the sample mean,
is the value tested at the null hypothesis,
is the standard deviation and n is the size of the sample.
Null hypothesis:
Tests
, which means that ![\mu = 5](https://tex.z-dn.net/?f=%5Cmu%20%3D%205)
In a sample of 8 students, he found the average time was 5.55 minutes and the standard deviation was 0.75 minutes.
This means, respectively, that ![n = 8, \mu = 5.55, \sigma = 0.75](https://tex.z-dn.net/?f=n%20%3D%208%2C%20%5Cmu%20%3D%205.55%2C%20%5Csigma%20%3D%200.75)
Test statistic:
The test statistic is:
![z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
![z = \frac{5.55 - 5}{\frac{0.75}{\sqrt{8}}}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B5.55%20-%205%7D%7B%5Cfrac%7B0.75%7D%7B%5Csqrt%7B8%7D%7D%7D)
![z = 2.07](https://tex.z-dn.net/?f=z%20%3D%202.07)
Pvalue:
Since we are testing if the mean is different from a value, and z is positive. The pvalue is two multiplied by 1 subtracted by the pvalue of z = 2.07.
z = 2.07 has a pvalue of 0.9808
2*(1 - 0.9808) = 2*(0.0192) = 0.0384
Decision:
0.0384 < 0.05, which means that we can conclude that the mean amount of time that college students spend in the shower is significantly different from 5 minutes.